It is widely held that certainty about the real world is a failed historical enterprise (that is, beyond deductive truths, tautology, etc.).[1] This is in large part due to the power of David Hume's problem of induction. Physicist Carlo Rovelli adds that certainty, in real life, is useless or often damaging (the idea is that "total security from error" is impossible in practice, and a complete "lack of doubt" is undesirable).[2]

Pyrrho is credited as being the first Skeptic philosopher. The main principle of Pyrrho's thought is expressed by the word acatalepsia, which denotes the ability to withhold assent from doctrines regarding the truth of things in their own nature; against every statement its contradiction may be advanced with equal justification. Secondly, it is necessary in view of this fact to preserve an attitude of intellectual suspense, or, as Timon expressed it, no assertion can be known to be better than another.

Al-Ghazali was a professor of philosophy in the 11th century. His book titled The Incoherence of the Philosophers marks a major turn in Islamic epistemology, as Ghazali effectively discovered philosophical skepticism that would not be commonly seen in the West until Averroes, René Descartes, George Berkeley and David Hume. He described the necessity of proving the validity of reason—independently from reason. He attempted this and failed. The doubt that he introduced to his foundation of knowledge could not be reconciled using philosophy. Taking this very seriously, he resigned from his post at the university, and suffered serious psychosomatic illness. It was not until he became a religious sufi that he found a solution to his philosophical problems, which are based on Islamic religion; this encounter with skepticism led Ghazali to embrace a form of theological occasionalism, or the belief that all causal events and interactions are not the product of material conjunctions but rather the immediate and present will of God.

Averroes was a defender of Aristotelian philosophy against Ash'ari theologians led by Al-Ghazali. Averroes' philosophy was considered controversial in Muslim circles.[3] Averroes had a greater impact on Western European circles and he has been described as the "founding father of secular thought in Western Europe".

Descartes' Meditations on First Philosophy is a book in which Descartes first discards all belief in things which are not absolutely certain, and then tries to establish what can be known for sure. Although the phrase "Cogito, ergo sum" is often attributed to Descartes' Meditations on First Philosophy, it is actually put forward in his Discourse on Method. Due to the implications of inferring the conclusion within the predicate, however, he changed the argument to "I think, I exist"; this then became his first certainty.

On Certainty is a series of notes made by Ludwig Wittgenstein just prior to his death. The main theme of the work is that context plays a role in epistemology. Wittgenstein asserts an anti-foundationalist message throughout the work: that every claim can be doubted but certainty is possible in a framework. "The function [propositions] serve in language is to serve as a kind of framework within which empirical propositions can make sense".[4]

Physicist Lawrence M. Krauss suggests that identifying degrees of certainty is under-appreciated in various domains, including policy making and the understanding of science. This is because different goals require different degrees of certainty—and politicians are not always aware of (or do not make it clear) how much certainty we are working with.[5]

Alternatively, one might use the legal degrees of certainty. These standards of evidence ascend as follows: no credible evidence, some credible evidence, a preponderance of evidence, clear and convincing evidence, beyond reasonable doubt, and beyond any shadow of a doubt (i.e. undoubtable—recognized as an impossible standard to meet—which serves only to terminate the list).

The foundational crisis of mathematics was the early 20th century's term for the search for proper foundations of mathematics.

After several schools of the philosophy of mathematics ran into difficulties one after the other in the 20th century, the assumption that mathematics had any foundation that could be stated within mathematics itself began to be heavily challenged.

Various schools of thought on the right approach to the foundations of mathematics were fiercely opposing each other. The leading school was that of the formalist approach, of which David Hilbert was the foremost proponent, culminating in what is known as Hilbert's program, which sought to ground mathematics on a small basis of a formal system proved sound by metamathematicalfinitistic means. The main opponent was the intuitionist school, led by L.E.J. Brouwer, which resolutely discarded formalism as a meaningless game with symbols.[citation needed] The fight was acrimonious. In 1920 Hilbert succeeded in having Brouwer, whom he considered a threat to mathematics, removed from the editorial board of Mathematische Annalen, the leading mathematical journal of the time.

Gödel's incompleteness theorems, proved in 1931, showed that essential aspects of Hilbert's program could not be attained. In Gödel's first result he showed how to construct, for any sufficiently powerful and consistent finitely axiomatizable system—such as necessary to axiomatize the elementary theory of arithmetic—a statement that can be shown to be true, but that does not follow from the rules of the system. It thus became clear that the notion of mathematical truth can not be reduced to a purely formal system as envisaged in Hilbert's program. In a next result Gödel showed that such a system was not powerful enough for proving its own consistency, let alone that a simpler system could do the job. This dealt a final blow to the heart of Hilbert's program, the hope that consistency could be established by finitistic means (it was never made clear exactly what axioms were the "finitistic" ones, but whatever axiomatic system was being referred to, it was a weaker system than the system whose consistency it was supposed to prove). Meanwhile, the intuitionistic school had failed to attract adherents among working mathematicians, and floundered due to the difficulties of doing mathematics under the constraint of constructivism.

In a sense, the crisis has not been resolved, but faded away: most mathematicians either do not work from axiomatic systems, or if they do, do not doubt the consistency of Zermelo–Fraenkel set theory, generally their preferred axiomatic system. In most of mathematics as it is practiced, the various logical paradoxes never played a role anyway, and in those branches in which they do (such as logic and category theory), they may be avoided.